This tag is for questions seeking external references (books, articles, etc.) about a particular subject. Please do not use this as the only tag for a question.

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3
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2answers
582 views

Which undergraduate math book cover topics about polynomials?

In China, there's a course called “Advanced Algebra," the first chapter of this course is all about polynomials, I wonder which undergraduate math books cover this topic? The details are below, sorry ...
6
votes
1answer
373 views

Measure on a separable Hilbert space

Let $H$ be a real separable Hilbert space. Is it true that there exist a probability space $(\Omega, \mu)$ and a measurable function $\pi\colon \Omega \to H$ such that for any $h \in H$ we have $$ ...
2
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0answers
163 views

Relations between elliptic curves and topological quantum field theory

I heard that there are relations between elliptic curves and topological quantum field theory (TQFT). I googled and found that something called "elliptic genus" might be the key word to relate these ...
9
votes
1answer
137 views

Proof of a basic $AC_\omega$ equivalence

On Wikipedia it is mentioned that "... in order to prove that every accumulation point $x$ of a set $S\subseteq \mathbf R$ is the limit of some sequence of elements of $S\setminus \{x\}$, one uses (a ...
2
votes
0answers
55 views

characteristic-$p$-type groups, and the Borel-Tits theorem for $PSL(V)$

If $G$ is a finite group and $F^*(G)$ is the generalized Fitting subgroup we say that $G$ has characteristic $p$ ($p$ is a prime that divides $|G|$) if $$F^*(G)=O_p(G)$$ Moreover $G$ is a ...
9
votes
3answers
714 views

Is $C([0,1])$ a “subset” of $L^\infty([0,1])$?

This is motivated from an exercise in real analysis: Prove that $C([0,1])$ is not dense in $L^\infty([0,1])$. My first question is how $C([0,1])$ is identified as a subset of $L^\infty([0,1])$? ...
3
votes
1answer
143 views

Moduli Spaces of Higher Dimensional Complex Tori

I know that the space of all complex 1-tori (elliptic curves) is modeled by $SL(2, \mathbb{R})$ acting on the upper half plane. There are many explicit formulas for this action. Similarly, I have ...
9
votes
3answers
614 views

Being ready to study calculus

Some background: I have a degree in computer science, but the math was limited and this was 10 years ago. High school was way before that. A year ago I relearnt algebra (factoring, solving linear ...
4
votes
2answers
99 views

Who defined $P$-names?

On reading Cohen's "Set Theory and the Continuum Hypothesis" it occurred to me that it might not have been Cohen himself who first defined $P$-names. In his book on page 113 he defines what he calls a ...
4
votes
2answers
115 views

The Dinitz problem

I would like to ask if someone knows about good books or online articles about The Dinitz problem or maybe someone can explain the problem a little. Consider $n^2$ cells arranged in an $( n \times ...
1
vote
2answers
269 views

Introductory Level Books for Graph Theory

Can anybody please suggest some good introductory level text books on Graph Theory ? Preferably those which don't really require a great pre-requisite background on discrete mathematics, but rather ...
13
votes
1answer
711 views

Real analysis textbok that develops the subject in a self-motivated, coherent fashion?

Well, it seems as though I just failed my analysis prelim for the second time... I have one more try in about $5$ months. I'm failing to build up a framework for how to think about analysis problems. ...
2
votes
1answer
95 views

Reference request: Construction of $M_{1,0}$

Does anyone know a reference for the construction of the (Artin) stack $M_{1,0}$ and a result about the corresponding coarse moduli space? In Deligne-Mumford they construct $M_{g,0}$ when $g\geq 2$ ...
2
votes
0answers
71 views

Categories of sheaves of finite homological dimension revisited

This is the question When is the category of (quasi-coherent) sheaves of finite homological dimension? revisited. I made some mistakes: I posted the question before I sign in, I tried to make it more ...
2
votes
3answers
200 views

I'm researching about geometry non-Euclidean [closed]

I'm researching non-Euclidean geometry. Now I am looking for a good source for it. Please suggest me something.Thanks.
3
votes
1answer
106 views

When is the category of (quasi-coherent) sheaves of finite homological dimension?

Let say from the beginning that my background is category of modules over a ring. So I know that if we take a given (nice) scheme $X$, then category of sheaves on $X$ is Grothendieck, so it must have ...
7
votes
1answer
111 views

Can we really understand $R$ by studying $R$-modules? [duplicate]

According to Algebra: Chapter 0, the category $R\operatorname{-Mod}$ reveals a lot about $R$. However after completing the first eight chapters, I still found no examples where this happens. Can ...
2
votes
0answers
49 views

Comparing the mean to the standard deviation

Let $X_1,X_2, \ldots ,X_n$ be i.i.d. random variables with normal distribution ${\cal N}(\mu,\sigma)$. Let $$ M=\frac{X_1+X_2+ \ldots +X_n}{n}, \ D=\sqrt{\sum_{k=1}^n (X_k-M)^2},\ Y=\frac{X_1+X_2+ ...
10
votes
4answers
807 views

A Math book with an inspiring ethos?

I was for some time curious about William Feller's probability tract (first volume); luckily, I could lay my hands on it recently and I find it of super qualities. It provides a complete exposition of ...
1
vote
2answers
58 views

Name for a path with least number of vertices.(Graph Geodesic)

What is the name given to a path between to points that is of length equal to the distance from those two pints. Sorry to bother you but the definition was on a book I no longer have.
2
votes
1answer
85 views

Looking for a paper of A.bella in 1987

Looking for a paper: Bella A. Remarks on the metrizability degree[J]. Boll. Un. Mat. Ital. A (7), 1987, 1(3): 391-396. Could you help me?
7
votes
0answers
197 views

Citation for subset complement result

Let $S = \{s_1, \ldots, s_n\} \subset \{1, \ldots, 2n\}$. Consider two operations on $S$, unfortunately both called complement in different setting: let $A(S) = \{1, \ldots, 2n\} \setminus S$ (set ...
7
votes
0answers
161 views

number of zeros of complex waves

Does anybody know about any type of methods how to calucalte/estimate the number of the zeros of complex waves (periodic functions as superposition of many harmonic waves) within a given period [0,x] ...
2
votes
0answers
168 views

Books for Practice Problems

I'm not asking for the solution to the question posted below, but instead for references to books where I can find similar questions. I'd be grateful for any advice you can give. Thank you.
3
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0answers
84 views

Equivariant homotopy equivalence of based loop group

Consider a compact, connected, simply connected Lie group $G$ and consider $S^1$ as an additive group. Let $\Omega G = \{ \gamma: S^1 \to G: \gamma(0) = e_G\}$ be the corresponding based loop group of ...
3
votes
2answers
199 views

PA with successor not being a function

The Peano axioms refer to the successor function $s$. What happens if we make $s$ non-deterministic, that is, when some natural numbers are allowed to have more than one successor? The resulting ...
1
vote
1answer
337 views

Definition of induced representation by tensor product

Suppose there is a finite group $G$ with a subgroup $H$ an some field $K$. If one has a representation of the subgroup, one can construct the induced representation $\rho:=Ind_H^G$ according to ...
2
votes
0answers
96 views

smoothness structure on a set

I'm reading Milnor's 'characteristic classes', and in the first chapter he defines smoothness structure on a set M, which is confusing for me, as what follows:(these are not Milnor's words so please ...
4
votes
1answer
76 views

Every factor of product topology $X$, is homeomorphic to a retract of $X$.

How can I prove this theorem? Let $X$ is the topological product of some family $\mathcal A$ of spaces, then every factor is homeomorphic to a retract of $X$.
1
vote
2answers
272 views

Reference on Polynomial Chaos

I need to understand the basics of "Polynomial Chaos" (http://en.wikipedia.org/wiki/Polynomial_chaos), and I'm having trouble finding a good reference on it. I'm looking for something rigorous enough ...
0
votes
1answer
48 views

$F:\bf C\to\bf D$ a functor with a right adjoint $G$ and $\bf S$ a full subcat of $\bf C$: When does the inclusion have a right adjoint?

Suppose a functor $F:\bf C\to\bf D$ has a right adjoint $G$, let $\bf S$ be a full subcategory of $\bf C$, and denote by $I$ the inclusion of $\bf S$ into $\bf C$. What are non-trivial assumptions ...
6
votes
1answer
234 views

Is there a rigorous theory of context, whereby sets can gain additional structure within a context?

Consider sets $G$ and $H$ and a function $f : G \rightarrow H$. So far, it doesn't really make sense to ask whether $G$ and $H$ are groups (technically, the answer is "no, they're not groups"), and ...
2
votes
1answer
204 views

Proof of an elliptic equation.

I'd like to see a proof of the theorem Theorem:Let $u \in H^1(B_1)$ a weak solution of \begin{equation} - \operatorname{div}(a_{ij}(x)\nabla u(x)) = 0 \quad \text{in} \quad B_1 \end{equation} ...
18
votes
1answer
390 views

References to integrals of the form $\int_{0}^{1} \left( \frac{1}{\log x}+\frac{1}{1-x} \right)^{m} \, dx$

While extending my calculation techniques, with aid of Mathematica, I found that \begin{align*} \int_{0}^{1}\left( \frac{1}{\log x} + \frac{1}{1-x} \right)^{3} \, dx &= -6 \zeta '(-1) ...
1
vote
1answer
195 views

Where can I learn about solving Big-Oh problems that are written in algebra? [duplicate]

Where can I learn about solving Big-Oh problems that are written in algebra? Such as this $$\sum_{i=1}^{n} (3i + 2n) = O(n^2)$$
11
votes
1answer
708 views

Book recommendations for self-study at the level of 3rd-4th year undergraduate

I have only recently discovered an interested in mathematics and I could only take a year off work to be back at school. Needless to say, for financial reasons (couple of mortgages) I will need to ...
2
votes
1answer
200 views

Math book with horsemen tessellation cover

I am looking for a math text I read when I was younger that had MC Escher's 'horsemen' tessellation on the cover. One of the Horsemen was colored red and the others in grey and white. As I remember, ...
7
votes
1answer
418 views

Sylow $p$-Subgroups of Classical Groups over $\mathbb{F}_p$

Let $p$ be a prime, and let $G$ be any of the finite classical groups $SL_n(\mathbb{F}_p)$, $O_n(\mathbb{F}_p)$, or $SP_n(\mathbb{F}_p)$. Let $P$ be a Sylow $p$-subgroup of $G$. What is $P$ as a ...
3
votes
0answers
132 views

Quotation and reference requested

I am looking for a quotation I once read, and I think it was by a famous mathematician. It is in the context of the balance between reading and writing mathematics, and said something like "I enjoy ...
3
votes
2answers
953 views

Ring theory reference books

I am studying ring theory in this semester. I am new to this theory. Hence, I would like to have some recommendations on what books should be used for ring theory(beginner). If possible, I would like ...
16
votes
1answer
491 views

Is there an axiomatic approach to ordinal arithmetic?

I've always wondered, is there an axiomatic approach to the arithmetic of ordinal numbers? If so, I imagine it would be on par with set theory in terms of its proof-theoretic strength.
3
votes
1answer
326 views

Where to start for studying geometry and what way should I follow after the first step?

I'm reading Gruenbaum's Tilings and Patterns: It's curious that almost all aspects of geometry relevant to the "man in the street" are ignored by our educational systems. Geometry has been almost ...
8
votes
1answer
210 views

why do most finite groups of order 128 resemble (at a distance) the elementary abelian group?

As a result of this previous question, I made the following video: Cayley Tables of All Groups of Order 128, and what is striking is that most of them, if you squint, kind of resemble the elementary ...
9
votes
2answers
164 views

Determining measures by integrals

What classes of functions are sufficient to determine whether two measures are equal? If $$\int_{R^d} f d\mu =\int_{R^d} f d\nu $$ for some functions $f$, when can we say that $\mu=\nu$? Obviously, ...
2
votes
0answers
232 views

A difficult, concise, and applicable complex analysis book.

I am currently going through Spivak's Calculus on Manifolds. I love the concision (only around 150 pages), and the problems are at just the right level for me (although I'd still be very happy if they ...
6
votes
3answers
223 views

Which books can I study (learn) better the topics about convergence in $\Bbb R^{n}$ and euclidean space?

I have question. Which books can I study (learn) better the topics about convergence in $\Bbb R^{n}$ and euclidean space ? Please can you give me an advice some book names? Thank you!
23
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5answers
481 views

Fractals reference

I want to present an elementary lecture about Fractals in the Nature. So, I am searching open or online references with good pictures like the following one: I prepared a good program that makes ...
6
votes
2answers
447 views

Rigour vs intuition

Researcher David Tall has written in chapter one of Advanced Mathematical Thinking that ...
4
votes
2answers
166 views

Subring of $R$ with fraction field $=$ Frac $R$

Let $R$ be an integral domain with fraction field $K$. Of course all the overrings of $R$ share the same fraction field $K$ (by an overring I mean a subring $S\subset K$ containing $R$ as a subring). ...
22
votes
1answer
854 views

A Combinatorial Proof of Dixon's Identity

Dixon's Identity states: $$ \sum_{k} (-1)^k\binom {a+b}{b+k}\binom{b+c}{c+k}\binom{c+a}{a+k} = \binom{a+b+c} {a,b,c}$$ A bit of history: The case $a=b=c$ was proved by Dixon in 1891 using ...